# Sample path large deviations and concentration

**Laplacian**particle interaction,

**Gaussian**potential with

**Wetting**.

We take our measure to be of the form

with boundary conditions imposed at .

Define the space

and the rate function

where and

We then obtain the following:

**Theorem:**

*The empirical profiles under the measures satisfy an LDP on the space with rate and rate function*.

Note that this theorem is for the Dirichlet boundary, with Gaussian potential. The wetting in this theorem imposes difficulties as our minimisers will have to change if there is a large initial downward gradient with a starting point near the origin. A few examples of how wetting changes minimisers is shown below. The orange lines represent a typical minimiser without wetting, and its corresponding minimiser with wetting is shown in blue.

The proof of this theorem involves a technical lemma which we state without proof

**Lemma:**

*There exists a coupling of random variables constructed on a common probability space such that where is the macroscopic height variable distributed according to the measure , with boundary .*

The main difficulty in the proof of the theorem is when the boundary values are zero. This introduces a technical problem that open sets in the space of continuous fuctions consistent with these boundary values contan functions which take negative values. The key lemma is used in the proof to 'lift' our boundary values away from zero and overcome this difficulty.

**free boundary case**, with

**general**potential V.